Shaft torque in rotating machinery is an important quantity needed for the design and monitoring of a system. In the past, the torque was calculated using approximate methods on the basis of rough measurement of the produced or consumed power. These methods provided approximate estimates of the average torque value under constant pump conditions, but were insufficient to provide the local variation of the torque along the shaft (i.e., torsional wave propagation) during transient conditions, e.g., when the load, torque or speed of rotation of the shaft is changed. Recently, instruments such as torque meters have been used to measure the shaft torque.
The majority of conventional torque meters use either of two principal methods for data collection. One type is based on strain gauges placed on the shaft with data collection through slip rings. The other type uses strain gauges on the shaft in conjunction with radio or magnetic transmitters for the data collection. Both of these methods entail significant alteration of the shaft and require open space on the shaft which is normally unavailable. Some torque meters require cutting of the shaft and insertion of the transducers. Electrical noise can be induced in the data transmission, leading to unacceptable errors. The insertion of mechanical sensing and data collection may itself reduce the useful life of the rotating part due to additional weight, misalignment and wear.
Therefore, a need exists for a less complicated, more accurate method of measuring the torque in rotating shafts which does not need a long free span and can be applied at different locations along the shaft.
Torsional vibrations in shafts of circular cross section are described by methods that are known in continuum mechanics. In particular, the angle .theta. through which a general cross section rotates about its equilibrium position obeys the differential equation: ##EQU1## where J(z) is the polar moment of inertia of the circular shaft (either hollow or solid), E.sub.s is the shear modulus, .rho. is the density of the shaft, and g.sub.O is the acceleration of gravity. The torque T is related to the angular displacement by: ##EQU2## Therefore, measurement of the angular displacement variation along the shaft, in general, implies deduction of the torque in the shaft.
For shafts which have uniform or slowly varying properties, the bracketed term in the above wave equation is small and can be neglected. This is the simplest case, with solutions that are traveling waves or standing waves (a superposition of two traveling waves). Such solutions can be expressed mathematically as: ##EQU3## Therefore, the vibrational torque is: EQU T(z,t)=-ik.theta..sub.O E.sub.s J(z) (e.sup.i(.omega.t-kz) .+-.e.sup.-i(.omega.t-kz))
whose real part is: EQU Re[T(z,t)]=2k.theta..sub.O E.sub.s J (z) sin(.omega.t-kz)
The amplitude coefficient .theta..sub.O is determined by the driving function amplitude operating on the end of the shaft.
These relationships imply that dynamic torque can be inferred from measurements of the properties of torsional wave propagation in the shaft. If the shaft is very nonuniform, then interpretation of the measurements is more complicated, yet feasible using series solutions of the more general differential equation. For example, a continuously variable shaft may possess a polar moment of inertia described by: ##EQU4##
The wave equation for this case becomes: ##EQU5## whose solution is known in terms of tabulated infinite series, called zero-order Hankel functions of the second kind: EQU .theta.(z,t)=.theta..sub.O H.sup.(2).sub.O (kz)e.sup.i.omega.t
If a time-varying torque of amplitude T.sub.O is applied at z=z.sub.O, the amplitude of the angular displacement is determined by: EQU T(z-z.sub.O,t)=T.sub.O e.sup.i.omega.t =kE.sub.s J.sub.O .theta..sub.O H.sup.(2).sub.1 (kz.sub.O)e.sup.i.omega.t
from which we find: ##EQU6## The torque in the shaft is then: ##EQU7## For short-wavelength vibrations, the Hankel function becomes: ##EQU8## so the torque approaches a phase-shifted traveling wave propagating toward positive z: ##EQU9## EQU z.sub.O .ltoreq.z.ltoreq.Z.sub.max ; kZ&gt;&gt;1
This result shows that torque is measurable as a traveling wave, even though the shaft is nonuniform.